Exponential Sum - Types of Exponential Sum

Types of Exponential Sum

Many types of sums are used in formulating particular problems; applications require usually a reduction to some known type, often by ingenious manipulations. Partial summation can be used to remove coefficients a_n, in many cases.

A basic distinction is between a complete exponential sum, which is typically a sum over all residue classes modulo some integer N (or more general finite ring), and an incomplete exponential sum where the range of summation is restricted by some inequality. Examples of complete exponential sums are Gauss sums and Kloosterman sums; these are in some sense finite field or finite ring analogues of the gamma function and some sort of Bessel function, respectively, and have many 'structural' properties. An example of an incomplete sum is the partial sum of the quadratic Gauss sum (indeed, the case investigated by Gauss). Here there are good estimates for sums over shorter ranges than the whole set of residue classes, because, in geometric terms, the partial sums approximate a Cornu spiral; this implies massive cancellation.

Auxiliary types of sums occur in the theory, for example character sums; going back to Harold Davenport's thesis. The Weil conjectures had major applications to complete sums with domain restricted by polynomial conditions (i.e., along an algebraic variety over a finite field).

One of the most general types of exponential sum is the Weyl sum, with exponents 2πif(n) where f is a fairly general real-valued smooth function. These are the sums implicated in the distribution of the values

ƒ(n) modulo 1,

according to Weyl's equidistribution criterion. A basic advance was Weyl's inequality for such sums, for polynomial f.

There is a general theory of exponent pairs, which formulates estimates. An important case is where f is logarithmic, in relation with the Riemann zeta function. See also equidistribution theorem.